Pressure and Flow of Exponentially Self-Correlated Active Particles. Sandford, C., Grosberg, A., Y., & Joanny, J. 2018.

Paper Website abstract bibtex

Paper Website abstract bibtex

Microscopic swimming particles, which dissipate energy to execute persistent directed motion, are a classic example of a non-equilibrium system. We investigate the non-interacting Ornstein– Uhlenbeck Particle (OUP), which is propelled through a viscous medium by a force which is cor-related over a finite time. We obtain an exact expression for the steady state phase-space density of a single OUP confined by a quadratic potential, and use the result to explore more complex ge-ometries, both through analytical approximations and numerical simulations. In a " Casimir " -style setup involving two narrowly-spaced walls, we describe a particle-trapping phenomenon, which leads to a repulsive effective interaction between the walls; while in a two-dimensional annulus geometry, we observe net stresses which resemble the Laplace pressure. Introduction. Recent investigation of " swimming " particles has provided many new insights into non-equilibrium phenomena. These swimmers exhibit a per-sistent Brownian motion, which violates detailed balance and the fluctuation-dissipation theorem, and results in a range of behaviours not observed in passive systems [1–5]. An " Ornstein–Uhlenbeck Particle " (OUP) swimmer is driven by a combination of a memory-less friction, and an exponentially correlated propulsion force with finite correlation time τ . This model has received significant attention, and some approximate methods have been pro-posed to study their steady state densities, such as the " Unified Coloured Noise Approximation " [6, 7] or per-turbative expansions close to equilibrium [8, 9]. In this paper we start with a simple exactly solv-able model of an OUP confined in a one dimensional harmonic potential, and discuss the crossover from an energy-equipartition dominated regime close to equilib-rium, to a force-balance dominated regime far from equi-librium. We use the results to interpret simulation data on more subtle OUP interactions with external poten-tials, including flows generated by asymmetric potentials, attractive and repulsive Casimir forces and Laplace-like pressure on a curved surface. Consider an OUP moving under an external force f (x) arising from a potential U (x), f = −∇U . In one dimen-sion (easily generalised to higher dimensions), the mi-croscopic equation of motion for the OUP's coordinate x(t) is the Langevin equation in which the propulsion force η(t) plays the role of a coloured noise and has ex-ponential correlations with a finite relaxation time τ . To treat this problem, we imagine that fluctuations of η(t) itself are governed by a hidden white noise variable ξ(t), such that the system as a whole is described by coupled Langevin equations:

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These swimmers exhibit a per-sistent Brownian motion, which violates detailed balance and the fluctuation-dissipation theorem, and results in a range of behaviours not observed in passive systems [1–5]. An \" Ornstein–Uhlenbeck Particle \" (OUP) swimmer is driven by a combination of a memory-less friction, and an exponentially correlated propulsion force with finite correlation time τ . This model has received significant attention, and some approximate methods have been pro-posed to study their steady state densities, such as the \" Unified Coloured Noise Approximation \" [6, 7] or per-turbative expansions close to equilibrium [8, 9]. In this paper we start with a simple exactly solv-able model of an OUP confined in a one dimensional harmonic potential, and discuss the crossover from an energy-equipartition dominated regime close to equilib-rium, to a force-balance dominated regime far from equi-librium. We use the results to interpret simulation data on more subtle OUP interactions with external poten-tials, including flows generated by asymmetric potentials, attractive and repulsive Casimir forces and Laplace-like pressure on a curved surface. Consider an OUP moving under an external force f (x) arising from a potential U (x), f = −∇U . In one dimen-sion (easily generalised to higher dimensions), the mi-croscopic equation of motion for the OUP's coordinate x(t) is the Langevin equation in which the propulsion force η(t) plays the role of a coloured noise and has ex-ponential correlations with a finite relaxation time τ . 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